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American Journal of Physics

Completeness of the energy eigenstates for a delta function potential

William C. Damert

Published Jun 1, 1975• 14 citations
Publisher

Abstract

By explicit calculation it is shown that the eigenfunctions of a particular Hamiltonian are complete. The Hamiltonian considered is that for a particle moving in one dimension with a Dirac delta function potential.
DOI
10.1119/1.9796

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Abstract The orthonormality and completeness of the energy eigenfunctions solution for the Schrödinger equation are usually hard to prove explicitly. There are few examples treated in the literature that give an explicit demonstration of these properties. In this work, we are going to orthonormalize the energy eigenfunctions of the solutions of the Schrödinger equation with contact potential given by the Dirac delta derivative potential λδ′(x), where λ is the coupling constant and prove that they are complete. The solutions for this problem include both continuum and discrete energy spectra. This makes it a perfect choice in the sense that it covers all possible cases of the solutions in quantum mechanics. The bound states solutions are normalized as usual, while the scattering states are normalized in continuum sense. The scattering solutions are shown to be orthogonal to each other and to the bound states. Using these orthonormal eigenfunctions an explicit demonstration of the completeness is given. It is found that both the bound and scattering solutions should be considered to ensure completeness. The calculation shows that the scattering state solutions are complete by themselves except for a certain value of the energy eigenvalue. This energy eigenvalue was shown to be precisely consistent with the energy eigenvalue of the bound state.
A pedagogical example is presented in which the completeness of the free state wavefunctions of a Hamiltonian can be explicitly checked. For certain values of the potential strength the free states prove to be complete. For other values of the potential strength the free states are shown to be incomplete and the extent of this incompleteness is shown to consist of precisely one function. Using the fact that the totality of all energy eigenfunctions must be complete, this single bound state wavefunction is calculated using the free state wavefunctions only.
There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
We delineate the scope of research on the completeness of eigenstates in quantum mechanics. Based on the limit of the potential function at infinity, the proof of completeness is divided into eight cases, and theoretical proofs or numerical simulations are provided for each case. We present the definition of orthonormalization for general free states and the solution to the normalization coefficients, as well as a general set of initial states, which simplifies and concretizes the proof of completeness. Additionally, we define the spectral function for continuous energy eigenvalues. By taking the spectral function as the original integral variable of the expansion function, the relationship between the measured probability amplitude and the expansion function is endowed with the physical meaning of coordinate-momentum transformation.